Abstract
This paper is devoted to the partial null controllability issue of parabolic linear systems with n equations. Given a bounded domain in R N, we study the effect of m localized controls in a nonempty open subset only controlling p components of the solution (p, m < n). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled 2x2 parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.
Highlights
Introduction and main resultsLet Ω be a bounded domain in RN (N ∈ N∗) with a C2-class boundary ∂Ω, ω be a nonempty open subset of Ω and T > 0
Y(0) = y0 in Ω, where y0 ∈ L2(Ω)n is the initial data, u ∈ L2(QT )m is the control and A ∈ L∞(QT ; L(Rn)) and B ∈ L∞(QT ; L(Rm, Rn)). In many fields such as chemistry, physics or biology it appeared relevant to study the controllability of such a system
In [10], the authors study a system of three semilinear heat equations which is a model coming from a mathematical description of the growth of brain tumors
Summary
To cite this version: Farid Ammar-Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control and Related Fields, AIMS, 2017, 6 (2), pp.Pages: 185 - 216. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Manuscript submitted to AIMS’ Journals Volume X, Number 0X, XX 200X doi:10.3934/xx.xx.xx.xx pp.
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